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Say a continuous random variable $X \in \mathbb{R}$ has a probability density function (p.d.f.).

Is it correct that,

change the probability measure $\Rightarrow$ define a new p.d.f. for X

?

In other words, is „changing the probability measure“ just a fancy way of saying „now assume that $X$ is distributed according to another p.d.f.?

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    $\begingroup$ While measure theory is important to probability theory (really, probability theory is a subset of measure theory), many people don’t really know it. It will help you get an appropriate answer if you edit your post to include your level of knowledge of measure theory and where you’re coming across the term “probability measure”. $\endgroup$ – Dave Oct 24 '20 at 12:36
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Yes, the classical PDF from beginning calculus-based probability is the “Radon-Nikodym derivative” of the probability measure with respect to Lebesgue measure. That more-or-less corresponds to taking the derivative of the CDF to get the PDF.

Even though the intro probability class makes it sound like we get CDFs by integrating densities, CDFs come directly from the probability measure, as $F_X(x) = P(X \le x)$ for the probability measure $P$. CDFs are, in some sense, more fundamental to formal, measure-theoretic probability, even if visualizing them is not as straightforward as a histogram or some other depiction of the density.

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